Let's get random together

Today we will talk about the Heinsenberg Uncertainty Principle. In the next post you will find its mathematical derivation.

Ok let’s start.

The Heisenberg Uncertainty Principle/Relation (herein HUP) is one of the most important concepts in QM. Even is it seems difficult, at least from a purely mathematical point of view, it has a nice physical explanation.
Of course, the HUP is easy to misinterpret due to the counterintuitive laws of quantum physics but with some easy examples it becomes feasible. In the following I will put an example (and you will also find the reference at the end of the post).

Suppose that we are dealing with very small objects having only two properties: color and hardness. Moreover, for the sake of clarity, let’s restrict the colours to black and white only.
Of course, also the hardness could be expressed in such discrete way, i.e., an object could be hard or soft with no intermediate degrees of fluffiness ^_^

Given an object A, it can be black and soft, white and soft, and so on.

Now, we do also have two boxes (what an insane amount of two-something); the first measures the color and the second measures the hardness.

Color Box

Hardness Box

Of course we can suppose that a composition of these two boxes allows to measure both the properties. Let’s see if this works.
The first box, the one which measures the colour works like this: we provide it one of our misterious objects in input and we observe; if the object comes from the upper exit, then our object is surely white, otherwise it is black.
Same thing for the hardness. In this way we can exactly know if the object is hard or soft by knowing the output position.
An important property of these boxes is the repeatability: when one of our small objects of a certain value of a given property is given in input into a box measuring the value of that property, it comes out with the original value for that measured property. I.e., if the object comes from the color box and it is white, if I give it in input to another color box it will come out white again. The same thing works for the hardness as well.

Repeatability for Color

Repeatability for Hardness

Pretty intuitive, this works also in our “classical” world.

Now that we know that these properties are persistent we can ask if color and hardness are correlated.
We can test it!
If we measure one property (let’s say color) of our small objects and we take all the examples with a single value of that property (the white ones) and we measure the other property (hardness), the value of the other property is found to be probabilistically evenly split. What does this mean? If our objects that are known to all be white have their hardness measured, half will be soft and half will be hard; similarly, if the objects that are soft have their color measured, half will be black and half will be white.
So… the measure of the value of one property gives no predictive power on the other property.

Example of Uncorrelation

Now that we know how to use these boxes, and how to compose them we can set an experiment slightly more sofisticated:

Experiment on Probabilities

We measure the hardness of a beam of small objects;

we consider only the soft objects and we send them through the color box;

objects exiting the color box are half black and half white;

we send the half that are black through another hardness box, and the previous measurements found that the electrons entering this box were soft and black;

since measurement of hardness (and color) is repeatable, these objects should always come out as soft and never as hard.

So, which is the probability that our objects are hard? Reasonably, this probability should be 0.

Wrong!
Half of the objects that were measured to only be soft are soft, and half now are black! The same thing works for any other pair of results from the first two boxes, if hardness and color were interchanged. Thus, the presence of the color box tampers with hardness, because without it, repeatability was ensured.
As a consequence of this bizarre fact, we cannot build a box to simultaneously (and reliably) measure both color and hardness since the measurement of one disturbs the other one and vice versa!

Simultaneous Measurement of Color and Hardness

Simultaneously measuring hardness and color is disallowed! The general statement of this is the uncertainty principle, which is the idea that some measurable physical properties of real systems are incompatible with each other in the way that has been described thus far. This nice example was inspired by the course of Allan Adams, Matthew Evans, and Barton Zwiebach. 8.04 Quantum Physics I, Spring 2013. (Massachusetts Institute of Technology: MIT OpenCourseWare).. If you have time check it out. I have slightly modified it not talking of electrons or photons in order to be as general as possible at the beginning.

Of course, this was an easy example to give you an idea of what are we talking about when we mention the Heisenberg Principle. Replace the color and hardness with two incompatible observables and we will do the trick.
I know it just looks magic but it is how nature works.

Next time I will introduce another nice example for the HUP, slightly more difficult, involving the famous double slit experiment.